Geometric significance of Toeplitz kernels

TL;DR

This paper explores the geometric aspects of Toeplitz kernels, linking the injectivity problem to geodesics in Grassmann manifolds and extending the analysis to Schatten ideals and commuting projections.

Contribution

It establishes a novel connection between Toeplitz operator injectivity and geodesic structures in Grassmann manifolds, expanding the geometric understanding of these operators.

Findings

Injectivity of Toeplitz operators relates to geodesics in Grassmann manifolds.

Extension of the geometric analysis to Schatten ideals.

Investigation of Toeplitz kernels in the context of commuting projections.

Abstract

Let $L^2$ be the Lebesgue space of square-integrable functions on the unit circle. We show that the injectivity problem for Toeplitz operators is linked to the existence of geodesics in the Grassmann manifold of $L^2$. We also investigate this connection in the context of restricted Grassmann manifolds associated to $p$-Schatten ideals and essentially commuting projections.